Answer to Question #311622 in Differential Equations for Deepak

Question

(d^3-5d^2+7d-3)y=0

Answer 1

The characteristic equation is

$D^3−5D^2+7D−3=0 ⟹D^3−D^2−4D^2+4D+3D−3=0 ⟹D^2(D−1)−4D(D−1)+3(D−1)=0$

$(D−1)(D^2−4D+3)=0 ⟹(D−1)(D−1)(D−3)=0 ⟹D=1∨D=1∨D=3$

Therefore the general solution is

$y=c_1e^x+c_2xe^x+c_3e^{3x}=(c_1+xc_2)e^x+c_3e^{3x }$